Dissipation-based entanglement via quantum Zeno dynamics and Rydberg antiblockade

A scheme is proposed for dissipative generation of maximally entanglement between two Rydberg atoms in the context of cavity QED. The spontaneous emission of atoms combined with quantum Zeno dynamics and the Rydberg antiblockade guarantees a unique steady solution of the master equation of the system, which just corresponds to the antisymmetric Bell state $|S\rangle$. The convergence rate can be accelerated by the ground-state blockade mechanism of Rydberg atoms. Meanwhile the effect of cavity decay is suppressed by the Zeno requirement, leading to a steady-state fidelity about $90\%$ as the single-atom cooperativity parameter $C\equiv g^2/\left(\kappa \gamma \right)=10$, and this restriction is further relaxed to $C=5.2$ once the quantum-jump-based feedback control is exploited.

Figure 1

(a) Schematic view of atomic-level configurations. The transition from the ground state $|g\rangle$ to the optical state $|p\rangle$ is resonantly coupled to the quantized cavity field with coupling strength $g$, and the transition between states $|p\rangle$ and $|e\rangle$ is driven by an optical pumping laser acting on the two atoms with Rabi frequency $\pm {\mathrm{\Omega }}_a$. In addition, there is a microwave field $\omega$ causing transition from the ground state $|g\rangle$ to the ground state $|e\rangle$, which is then pumped to the excited Rydberg state $|r\rangle$ by a classical field ${\mathrm{\Omega }}_b$, detuning $-\mathrm{\Delta }$. (b) The effective transitions for two atoms. The whole system works well in a subspace of zero occupation for the cavity mode due to the quantum Zeno dynamics. Under the coaction of the spontaneous emission of state $|D\rangle$, the optical pumping laser ${\mathrm{\Omega }}_a$, the microwave driving field $\omega$, and the Rydberg antiblockade interaction $\lambda$, the system finally will be stabilized into the singlet state $|S\rangle$ because of $\mathrm{\Gamma }\ll \gamma$.

Figure 2

The purity $P\left(t\right)=\mathrm{Tr}\left[{\rho }^2\left(\mathrm{t}\right)\right]$ is plotted as a function of time with initial states $|gg\rangle$ (black solid line), $|T\rangle$ (red dashed line), and $|ee\rangle$ (green dotted line), respectively. The spontaneous emission rate of the atom is assumed as $\gamma =0.1g$, and the other parameters for the calculation are chosen as (a) ${\mathrm{\Omega }}_a=0.1g,\omega =0.05g,{\mathrm{\Omega }}_c=0.5g,\mathrm{\Delta }=10g$; (b) ${\mathrm{\Omega }}_a=0.05g,\omega =0.025g,{\mathrm{\Omega }}_c=0.5g,\mathrm{\Delta }=20g$; (c) ${\mathrm{\Omega }}_a=0.05g,\omega =0.025g,{\mathrm{\Omega }}_c=5g,\mathrm{\Delta }=100g$; (d) ${\mathrm{\Omega }}_a=0.05g,\omega =0.025g,{\mathrm{\Omega }}_c=10g,\mathrm{\Delta }=200g$.

Figure 3

The dependence of the fidelity $F\left(t\right)=\langle S\left|\rho \right(t\left)\right|S\rangle$ on time is illustrated from the initial state $|gg\rangle$ for a perfect cavity and a leaky cavity, respectively. The other fixed parameters are set as $\omega =0.05g,{\mathrm{\Omega }}_c=0.5g,\mathrm{\Delta }=10g$ for the solid line and dashed line, and $\omega =0.025g,{\mathrm{\Omega }}_c=0.5g,\mathrm{\Delta }=20g$ for the dash-dotted line and dotted line. Compared with the case of ${\mathrm{\Omega }}_a=0.1g$, the fidelity is improved for a weaker driving field ${\mathrm{\Omega }}_a=0.05g$ at the cost of increasing the convergence time. The empty circles simulate the dynamics from the effective master equation corresponding to the dash-dotted line.

Figure 4

(a) Contour plot of steady-state fidelity versus atomic spontaneous emission $\gamma /g$ and cavity decay $\kappa /g$ under the given parameters: ${\mathrm{\Omega }}_a=0.01g,\omega =0.5{\mathrm{\Omega }}_a,{\mathrm{\Omega }}_c=g,\mathrm{\Delta }=20g,\mathrm{\Gamma }=0.001g$. The dash-dotted line is simulated as a fidelity around 90% corresponding to the single-atom cooperativity parameter $C=10$. (b) Contour plot of the enhanced fidelity for feedback-based steady state versus atomic spontaneous emission $\gamma /g$ and cavity decay $\kappa /g$. The feedback parameter is chosen as $\eta =0.5\pi$, and the other relevant parameters are set as ${\mathrm{\Omega }}_a=0.01g,\omega =0.5{\mathrm{\Omega }}_a,{\mathrm{\Omega }}_c=g,\mathrm{\Delta }=20g,\mathrm{\Gamma }=0.001g$, in order to obtain a 90% fidelity at a small value of $C=5.2$ (as shown by the dash-dotted line).

Figure 5

The steady-state fidelity versus the deviation $\delta /\mathrm{\Delta }$, where $\kappa =\gamma =0.1g$ is assumed and other parameters are the same as of Fig. 4.